Ruled Real Hypersurfaces in the Indefinite Complex Projective Space
aa r X i v : . [ m a t h . DG ] F e b Ruled Real Hypersurfaces in theIndefinite Complex Projective Space
Marilena Moruz , Miguel Ortega and Juan de Dios P´erez ∗ Al.I. Cuza University of Ia¸si, Faculty of MathematicsBd. Carol I, n. 11, 700506 Ia¸si, [email protected] Department of Geometry and Topology, Faculty of SciencesCampus de Fuentenueva, 18071 Granada, [email protected], [email protected] 23, 2021
Abstract
The main two families of real hypersurfaces in complex space forms are Hopf andruled. However, very little is known about real hypersurfaces in the indefinite complexprojective space C P np . In a previous work, Kimura and the second author introduced Hopfreal hypersurfaces in C P np . In this paper, ruled real hypersurfaces in the indefinite com-plex projective space are introduced, as those whose maximal holomorphic distributionis integrable, and such that the leaves are totally geodesic holomorphic hyperplanes. Adetailed description of the shape operator is computed, obtaining two main different fam-ilies. A method of construction is exhibited, by gluing in a suitable way totally geodesicholomorphic hyperplanes along a non-null curve. Next, the classification of all minimalruled real hypersurfaces is obtained, in terms of three main families of curves, namelygeodesics, totally real circles and a third case which is not a Frenet curve, but can beexplicitly computed. Four examples are described. Keywords:
Real hypersurface, indefinite complex projective space, ruled real hypersurface.
Primary 53B25, 53C50; Secondary 53C42, 53B30.
The study of real hypersurfaces in non-flat complex space forms has been strongly developedduring the last 50 years. Since a short list would not be fair, a good survey can be foundin Cecil and Ryan’s book [6]. Anyway, the main pillars of the theory of real hypersurfaces ∗ First author is supported by a grant of the Romanian Ministry of Education and Research, CNCS - UE-FISCDI, project number PN-III-P1-1.1-PD-2019-0253, within PNCDI III. Second and third authors partiallyfinanced by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund(ERDF), project MTM2016-78807-C2-1-P. Second author belongs to the Excellence Scientific Unit ’Science inthe Alhambra’ of Granada University, ref. UCE-PP2018-01.
1n complex projective space C P n are the following papers. First, Takagi, [19], classified thehomogeneous ones, obtaining a list of 6 examples. Second, Cecil and Ryan developed thefocal theory in [5], obtaining local characterizations of Hopf examples, namely, those whoseReeb vector field ξ = − J N is principal, where N is a unit normal vector field and J is thecomplex structure of C P n . Next, Kimura [8] characterized the Takagi’s examples as the onlyones which are Hopf and all of whose principal curvatures are constant. Similar studies forthe complex hyperbolic space were made, but we recommend again [6].In addition, Kimura also introduced in [9] a very important family of real hypersurfacesin the complex projective space, the so-called ruled ones. Indeed, if D denotes the maximalholomorphic distribution on M , given at each point x ∈ M by all the vectors orthogonal to ξ x , a ruled real hypersurface is such that D is integrable and its leaves are totally geodesiccomplex projective spaces C P n − in C P n . Among the works on ruled real hypersurfaces, werecall the one by Lohnherr and Reckziegel, [12], who proved that any ruled real hypersurfacecan be parametrized from a certain curve on C P n , among other results. Some recent resultsconcerning them are [10] and [17]. Summing up, both Hopf and ruled real hypersurfaces aretwo of the main families, because they appear in plenty of results.If we consider the indefinite complex projective space C P np of index 1 ≤ p ≤ n −
1, see [3] forits Kaehlerian structure, the beginning of the study of its non-degenerate real hypersurfaces M can be dated in 2015 by Anciaux and Panagiotidou, [2]. They obtained an almost contactmetric structure on M and proved a few basic results, like the non-existence of either totallyumbilical real hypersurfaces or real hypersurfaces with parallel shape operator in C P np . Theyalso posed a couple of problems.In [11], Kimura and the second author obtained several families of Hopf real hypersurfacesin C P np , which were named of type A + , A − , B , B + , B − and C because they are somehowsimilar to those in Takagi’s and Montiel’s list [13, 19]. They also showed an example of adegenerate (light-like) Hopf real hypersurface. This meant a positive answer to a questionasked by Anciaux and Panagiotidou. All cases A + , A − , B , B + and B − , were constructedas tubes of certain radius r over some holomorphic totally geodesic submanifolds of complexdimension m . Also, the subindex ± indicates the causal character of the unit normal, except B , which is timelike. Example C is similar to the horosphere in the complex hyperbolic space,so they are also called horospheres. Next, in the same paper, they also proved that an η -umbilical ( AX = λX + ǫµη ( X ) ξ , for any X tangent to M ) non-degenerate real hypersurfacein C P np must be either of type A + or A − or C . They also classified real hypersurfaceswhose Weingarten endomorphism is diagonalizable and satisfy that ξ is a Killing vector field,answering another question posed in [2].Since Hopf real hypersurfaces in C P np have been already introduced, the purpose of thepresent paper is to begin the study of ruled real hypersurfaces in C P np , in the following way:1. Section 2 is devoted to obtaining three kinds of axioms of planes in C P np that yieldthe existence of some families of totally geodesic submanifolds passing through a givenpoint x ∈ C P np and tangent to some vector subspaces of T x C P np . We also introduceFrenet curves in C P np .2. In Section 3, we define ruled real hypersurfaces in C P np as non-degenerate real hyper-surfaces whose maximal holomorphic distributions are integrable with totally geodesicleaves in the ambient space, obtaining a characterization of such real hypersurfaces interms of the shape operator. We also make a description of the shape operator of suchhypersurfaces. In Section 3.2 we introduce ruled hypersurface parametrizations in C P np M in C P np and each point q ∈ M , there exists a unique parametrization of M around q , centeredby an integral curve α of the Reeb vector field ξ . All this makes sense due to the axiomof planes for maximal holomorphic tangent planes, as in Section 2.1.3. Finally, in Section 4, Theorem 2 states the classification of minimal ruled real hyper-surfaces, obtaining three possibilities for the integral curves of ξ . Analyzing such threecases and using the axioms of planes in Section 2.1, we introduce several examples ofruled real hypersurfaces in C P np , which depend on the causal character of the curve.These curves can be either a geodesic, a totally real circle (as in [1]) contained in atotally real, totally geodesic surface, or a non-Frenet curve contained in a totally real,totally geodesic 3-submanifold. C P np Let C n +1 p be the Euclidean complex space endowed with the following Hermitian product andpseudo-Riemannian metric of index 2 p , z = ( z , . . . , z n +1 ), w = ( w , . . . , w n +1 ) ∈ C n +1 , g C ( z, w ) = − p X j =1 z j ¯ w j + n +1 X j = p +1 z j ¯ w j , g = Re( g C ) , (1)where ¯ w is the complex conjugate of w ∈ C n +1 . The natural complex structure will be denotedby J . As usual, we define the set S = { a ∈ C : a ¯ a = 1 } = { e iθ : θ ∈ R } . We consider thehyperquadric S n +12 p = { z ∈ C n +1 p : g ( z, z ) = 1 } , which is a semi-Riemannian manifold of index 2 p . We define the action and its correspondingquotient S × S n +12 p → S n +12 p , ( a, ( z , . . . , z n +1 )) ( az , . . . , az n +1 ) ,P i : S n +12 p → C P np = S n +12 p / ∼ . Let g be the metric on C P np such that P i becomes a semi-Riemannian submersion. Themanifold C P np is called the Indefinite Complex Projective Space . See [3] for details. We need1 ≤ p ≤ n − C P n with either a Riemannian or a negative definite metric. Let ˜ ∇ be its Levi-Civita connection. Then, C P np admits a complex structure J induced by P i , withRiemannian tensor¯ R ( X, Y ) Z = g ( Y, Z ) X − g ( X, Z ) Y + g ( J Y, Z ) J X − g ( J X, Z ) J Y + 2 g ( X, J Y ) J Z, (2)for any
X, Y, Z ∈ T M . Thus, C P np has constant holomorphic sectional curvature 4.By an abuse of notation, we will denote by J all the complex structures. It is importantto bear in mind that for q ∈ S n +12 p , T q S n +12 p = { X ∈ C n +1 p : g ( X, q ) = 0 } . x ∈ C P np , let q ∈ S n +12 p such that P i ( q ) = x. We have
P i : S n +12 p → C P np and dP i q : T q S n +12 p → T x C P np . (3)In general we will refer to dP i q as to the restriction dP i q : (Ker( dP i q )) ⊥ → T x C P np , (4)where Ker( dP i q ) = span { J ~q } , since this restriction is an isometry. Therefore, the map P i is a semi-Riemannian submersion, i.e.,
P i has maximal rank (each derivative map dP i issurjective) and dP i preserves lengths of horizontal vectors.Let D and ˆ ∇ be the Levi-Civita connections of C n +1 p and S n +12 p , respectively. Since theposition vector χ : S n +12 p → C n +1 p behaves as a spacelike, unit, normal vector field, the Gaussformula becomes D X Y = ˆ ∇ X Y − h X, Y i χ, (5)for any X, Y tangent to S n +12 p .The following 3 lemmatta state that C P np satisfies certain types of axioms of planes .Essentially, we show the existence of some families of totally geodesic submanifolds, passingthrough a given point and tangent to some tangent planes. The ideas underneath come fromthe fact that totally geodesic submanifolds of some model spaces embedded in flat R nm can becomputed by intersecting these models with linear subspaces. For more details, see [15]. Werecall the following definitions. In any complex manifold ( ¯ M , ¯ g, J ), a tangent plane at z ∈ ¯ M ,Π ⊂ T z ¯ M , dim Π ≥
2, is called holomorphic , respectively totally real , if J Π = Π, respectively J Π ⊥ Π. Accordingly, a submanifold S ⊂ ¯ M is holomorphic (resp. totally real ) when for any p ∈ S , T p S is holomorphic (resp. totally real). Lemma 1.
Let x ∈ C P np and consider a holomorphic, non-degenerate hyperplane Π ⊂ T x C P np of dimension dim R Π = 2 n − . The following hold:1. There exists a totally geodesic submanifold L in C P np such that x ∈ L and T x L = Π . Inaddition, L is isometric to C P n − t , where t = p if Π ⊥ is definite positive or t = p − if Π ⊥ is definite negative.2. Any two totally geodesic submanifolds L and L as above are linked by an isometry of C P np .Proof. Step 1: We start by pointing out the following totally geodesic embeddings: χ + : S n − p → S n +12 p , ( z , . . . , z n ) ( z , . . . , z n , ,χ − : S n − p − → S n +12 p , ( z , . . . , z n ) (0 , z , . . . , z n ) . It is quite clear that we obtain the following totally geodesic embeddings: C P n − p → C P np , C P n − p − → C P np . If we denote o = (0 , . . . , , ∈ S n − p , Q + = (0 , . . . , , , Q − = (1 , , . . . , ∈ C n +1 p , then T o S n − p = { X ∈ C np : Re( X n ) = 0 } , T ( o, S n +12 p = { X ∈ C n +1 p : Re( X n ) = 0 } , ( dχ + ) o ( X ) = ( X, , ∀ X ∈ T o S n − p , ( dχ + ) o (cid:0) T o S n − p (cid:1) = { Z ∈ C n +1 p : Re( Z n ) = 0 , Z n +1 = 0 } = Span { ( o, , Q + , J Q + } ⊥ . S n − p − , using Q − . Step 2:
Take x ∈ C P np , and a non-degenerate Π ⊂ T x C P np , such that J Π = Π, dim R Π =2 n −
2. Consider q ∈ S n +12 p such that P i ( q ) = x . There exists a unit η ∈ T x C P np such thatΠ ⊥ = Span { η, J η } . Let ˆ η be its horizontal lift via P i . Construct an R -orthonormal basis B = ( v , J v , . . . , v n +1 , J v n +1 ) in C n +1 p in the following way. If ˆ η is spacelike, then v n = q , v n +1 = ˆ η . If ˆ η is timelike, then v n = q , v = ˆ η . Define the matrix M such that its k th columnis v k . If det( M ) = −
1, we change a suitable v k by − v k . Since B is a real orthonormal basis, itis easy to check that M ∈ SU ( p, ¯ p ) = { M ∈ M n +1 ( C ) | ¯ M t I p, ¯ p M = I p, ¯ p , det M = 1 } , for ¯ p = n + 1 − p , where I p, ¯ p is the matrix I p, ¯ p = Diagonal( − , ( p . . ., − , , (¯ p . . ., f M : C n +1 p → C n +1 p can be restricted to S n +12 p as an isometry, satisfying f M ( o,
0) = q ,( df M ) ( o, Q ± = ˆ η . In addition, it can be projected to C P np , obtaining a holomorphic isometry f : C P np → C P np such that f ( P i ( o, x and f ∗ ( P i ∗ ( Q ± )) = η . Step 3:
The desired totally geodesic submanifold is L x = f ( P i ( S n − t )), for t = p if η isspacelike or t = p − η is timelike. This totally geodesic submanifold is unique at the point x , up to the chosen hyperplane. By composing two of the above isometries, any two totallygeodesic submanifolds as above will be linked by a holomorphic isometry.We provide a list of totally real, totally geodesic surfaces in C P np . In all cases, we denote ˆΣthe surface embedded in S n +12 p , and its projection Σ = P i ( ˆΣ), as in the following commutativediagram ˆΣ S n +12 p Σ C P npt.g. ˆ ϕP i P it.g. ϕ We will follow [15, p.110]. We also recall that a map f : ( ¯ M , ¯ g ) → ( ¯ M , ¯ g ) is an anti-isometry if for each p ∈ ¯ M and each u, v ∈ T p ¯ M , it holds g p ( u, v ) = − g f ( p ) ( f ∗ u, f ∗ v ). As a consequence,the sectional curvatures of 2-planes linked by f have different signs.As above, all embeddings in C P np are totally real and totally geodesic. R P ) For n ≥
3, 1 ≤ p ≤ n −
2, the round sphere ˆ S = { ( x, y, z ) ∈ R : x + y + z = 1 } ,and ˆ ϕ : ˆ S → S n +12 p , ( x, y, z ) (0 , . . . , , x, y, z ). By the classical quotient R P =ˆ S / {± I } , I the identity matrix of order 3, we get an embedding of the Riemannianreal projective plane ϕ : R P → C P np . H ) For n ≥
3, 2 ≤ p ≤ n −
1, let ˆ S = { ( x, y, z ) ∈ R : − x − y + z = 1 } , and ˆ ϕ :ˆ S → S n +12 p , ( x, y, z ) ( x, y, , . . . , , z ). Take the classical quotient H := ˆ S / {± I } .Then, we get ϕ : H → C P np . By Lemma 24 of [15, p.110], H is anti-isometric to thestandard real hyperbolic plane. S ) For n ≥
2, 1 ≤ p ≤ n −
1, take ˆ S = { ( x, y, z ) ∈ R : − x + y + z = 1 } . Itsuniversal covering is the de Sitter ϕ : ˆ S → S n +12 p , ( x, y, z ) ( x, , . . . , , y, z ), induces ϕ : S = ˆ S/ {± I } → C P np . Lemma 2.
Given x ∈ C P np , consider a non-degenerate, totally real, 2-plane Π ⊂ T x C P np . . There exists a non-degenerate, totally real, totally geodesic surface Σ ⊂ C P np such that x ∈ Σ and T x Σ = Π .2. Any two surfaces Σ and Σ as in item 1 are linked by an isometry of C P np if, and onlyif, they have one causal character among the previous cases R P , H and S .Proof. Since C P np has constant holomorphic sectional curvature +4, totally real, totallygeodesic submanifolds have constant sectional curvature +1. Then, we look for the modelspaces of surfaces with constant Gaussian curvature K = +1, with index 0, 1 and 2. Theseare described in the previous list. Finally, the proof can be finished by a similar technique toLemma 1.Next, we show the list of totally real, totally geodesic, 3-dimensional submanifolds in C P np with index 1 or 2. We will not repeat that all the embeddings will be totally real and totallygeodesic. The ideas are very similar to the previous list of surfaces.( B ) For n ≥
3, 1 ≤ p ≤ n −
2, take the de Sitter space ˆ ψ : dS = { p = ( x , x , x , x ) ∈ R : − x + x + x + x = 1 } → S n +12 p , p ( x , , . . . , , x , x , x ). This induces B = dS / {± I } → C P np , where I is the identity matrix of order 4. The index of B is 1.( B ) For n ≥
3, 2 ≤ p ≤ n −
1, take the hyperquadric Q = { p = ( x , x , x , x ) ∈ R : − x − x + x + x = 1 } → S n +12 p so we obtain B = Q / {± I } → C P np . The index is2. By Lemma 24 [15, p.110], B is locally anti-isometric to the anti-de Sitter Lemma 3.
Given x ∈ C P np , consider a non-degenerate, totally real 3-plane Π ⊂ T x C P np ofindex 1 or 2.1. There exists a non-degenerate, totally real, totally geodesic, 3-submanifold B ⊂ C P np such that x ∈ B and T x B = Π .2. Any two 3-submanifold B and B as in item 1 are linked by an isometry of C P np if,and only if, they have the same causal character, among the previous cases B and B .Proof. Since the index has to be 1 or 2, we consider the family of simply connected, 3-manifolds of constant sectional curvature of index 1 or 2. We can use [15, p.110] to constructthem. When the index is 1, then it is (locally) the de Sitter 3-space. When the index is 2,then it is (locally) Q . The previous list provides some model immersions.We finish the proof again as in Lemma 1.We will define next a Frenet curve of order r ≥ C P np . When r >
1, we are assumingthat the Frenet vectors involved in the Frenet system of equations are never lightlike. Givena curve α : I → C P np , we denote by D/ds the covariant derivative along it.
Definition 1 ( Frenet curves of order r ≥ in C P np ) . Given a unit curve α : I → C P np ,calling α ′ = E , we have: If α is a geodesic, then r = 1 , its Frenet vector is { E } and its renet curvature is κ = 0 . If α is not a geodesic, then DE ds = ε κ E ,DE ds = − ε κ E + ε κ E ,DE ds = − ε κ E + ε κ E , ... DE r ds = − ε r − κ r − E r − , where { E , . . . , E r } are orthonormal vectors along α with lengths g ( E i , E i ) = ε i = ± and κ , . . . , κ r − their Frenet curvatures. Together they describe the Frenet system of the Frenetcurve α of order r . If r = n , then the last vector E r = E n is chosen such that E , . . . , E n form a positive basis. All the Frenet curvatures κ k > , except if r = n , when κ n − might bepositive or negative. C P np For details, see [11]. Let M be a connected, non-degenerate, immersed real hypersurface in C P np . If N is a local unit normal vector field such that ε = g ( N, N ) = ±
1, we define the structure vector field on M as ξ = − J N . Clearly, g ( ξ, ξ ) = ε . Given X ∈ T M , the vector
J X might not be tangent to M . Then, we decompose it in its tangent and normal parts, namely J X = φX + εη ( X ) N, where φX is the tangential part, and η is the 1-form on M such that η ( X ) = g ( J X, N ) = g ( X, ξ ) . The set ( g, ξ, φ, η ) is called an almost contact metric structure, whose properties are η ( φX ) = 0 , φ X = − X + εη ( X ) ξ,g ( φX, φY ) = g ( J X − εη ( X ) N, J Y − εη ( Y ) N ) = g ( X, Y ) − εη ( X ) η ( Y ) , (6) g ( φX, Y ) + g ( X, φY ) = 0 , φξ = 0 , η ( ξ ) = ε, ( ∇ X φ ) Y = εg ( Y, ξ ) AX − εg ( AX, Y ) ξ, for any X, Y ∈ T M . Next, if ˜ ∇ and ∇ are the Levi-Civita connections of C P np and M ,respectively, we have the Gauss and Weingarten formulae:˜ ∇ X Y = ∇ X Y + εg ( AX, Y ) N, ˜ ∇ X N = − AX, (7)for any
X, Y ∈ T M , where A is the shape operator associated with N . Also, ∇ X ξ = φAX, (8)for any X ∈ T M . The Codazzi equation is( ∇ X A ) Y − ( ∇ Y A ) X = η ( X ) φY − η ( Y ) φX + 2 g ( X, φY ) ξ, (9)7or any X, Y ∈ T M . Let R be the curvature operator of M . Then, by using (2), (7) and thestructure Gauss equation [15, pag. 100], we obtain R ( X, Y ) Z = g ( Y, Z ) X − g ( X, Z ) Y + g ( φY, Z ) φX − g ( φX, Z ) φY − g ( φX, Y ) φZ + εg ( AY, Z ) AX − εg ( AX, Z ) AY, for any
X, Y, Z ∈ T M .We recall the following theorem from [11]. A real hypersurface M is called η -umbilical ifthere exist functions a, b ∈ C ∞ ( M ) such that AX = aX + εbη ( X ) ξ , for any X ∈ T M . Thedescriptions of the real hypersurfaces in the next theorem can be checked in detail in [11].
Theorem 1.
Let M be a connected, non-degenerate, oriented real hypersurface in C P np , n ≥ , such that it is η -umbilical. Then, M is locally congruent to one of the following realhypersurfaces:1. A real hypersurface of type A + , with m = q + 2 , q ≤ p ≤ m = q + 2 , µ = 2 cot(2 r ) and λ = cot( r ) , r ∈ (0 , π/ ;2. A real hypersurface of type A + , with m = n + q + 1 , ≤ q ≤ , µ = 2 cot(2 r ) and λ = − tan( r ) , r ∈ (0 , π/ ;3. A real hypersurface of type A − , with m = q + 2 , q ≤ p ≤ m = q + 2 , µ = 2 coth(2 r ) , r > and λ = coth( r ) ;4. A real hypersurface of type A − , with m = q + 2 , q ≤ p ≤ m = q + 2 , µ = 2 coth(2 r ) , r > and λ = tanh( r ) ;5. A real hypersurface of type C , also known as a horosphere. In this section we will first give the definition of ruled real hypersurfaces and present somebasic characterizations. We will determine the form of the shape operator and then we willdiscuss the parametrization of ruled real hypersurfaces.Let M be a hypersurface in C P np and let q ∈ M . The maximal holomorphic distributionis defined as D q := T q M ∩ J T q M = span { ξ q } ⊥ . Definition 2.
A non-degenerate real hypersurface M in C P np is ruled if, and only if, themaximal holomorphic distribution D is integrable with totally geodesic leaves in C P np . Lemma 4.
Let M be a non-degenerate real hypersurface. Then M is ruled if, and only if, g ( AX, Y ) = 0 , ∀ X, Y ∈ D .Proof.
Assume that M is ruled. Let L be a totally geodesic leaf of D in C P np . We call ∇ L its Levi-Civita connection. By Gauss equation, since L ⊂ C P np , for any X, Y ∈ T L , wehave ˜ ∇ X Y = ∇ LX Y , which proves that ˜ ∇ X Y ∈ T L . Moreover, since
D ⊂
T M ⊂ C P np and L ⊂ M ⊂ C P np , we have that ˜ ∇ X Y = ∇ X Y + εg ( AX, Y ) N. N is not a lightlike vector, we take the inner product of the above equation with N , andusing that N ⊥ D , T L ⊂ D , we obtain g ( ˜ ∇ X Y, N ) = g ( ∇ X Y, N ) + ε g ( AX, Y ), that is tosay, 0 = g ( AX, Y ).We assume now that g ( AX, Y ) = 0 , ∀ X, Y ∈ D . We easily obtain 0 = g ( AX, Y ) = − g ( ˜ ∇ X N, Y ) = g ( N, ˜ ∇ X Y ), for any X, Y ∈ D , which implies that ˜ ∇ X Y ∈ T M . We knowthat
J Y = φY ∈ D , for any Y ∈ D . It follows that 0 = g ( AX, φY ) = − g ( ˜ ∇ X N, J Y ) = g ( N, ˜ ∇ X J Y ) = g ( N, J ˜ ∇ X Y ) = g ( ξ, ˜ ∇ X Y ) . Therefore, ˜ ∇ X Y is tangent to M and orthogonalto ξ , which implies that ˜ ∇ X Y ∈ D . Similarly, we show that ˜ ∇ Y X ∈ D . Hence, for X, Y ∈ D ,[ X, Y ] = ˜ ∇ X Y − ˜ ∇ Y X ∈ D and so D is an integrable distribution. Next, let us see that theleaves of D are totally geodesic. Take L as one of them, that is, L is a submanifold of C P np such that for any p ∈ L , we have T p L = D p . Let ∇ L and σ be the Levi-Civita connectionon L and the second fundamental form of L in C P np . Therefore, for X, Y ∈ T L , the Gaussequation writes as ˜ ∇ X Y = ∇ LX Y + σ ( X, Y ) . In fact, for
X, Y ∈ Γ( T L ), we have that ∇ LX Y ∈ Γ( T L ) and ˜ ∇ X Y | q ∈ D q = T q L, for q ∈ L .It follows that σ ( X, Y ) = 0 and, hence, L is totally geodesic. Finally, by Lemma 1, the leavesare isometric to some C P n − t , where t = p if N is spacelike or t = p − N is timelike. From now on, M will denote a ruled real hypersurface in C P np , unless otherwise stated.Suppose there exists an open subset Ω in M on which the shape operator of M vanishes: A | Ω ≡
0. It implies that Ω is totally geodesic, but this leads to a contradiction as follows.For
X, Y ∈ T M , the Codazzi equation (9) implies that 0 = η ( Y ) φX − η ( X ) φY − g ( X, φY ) ξ and we obtain immediately that g ( X, φY ) = 0, for all
X, Y ∈ T M . We see that for a unitvector X ∈ D and Y = φX , we obtain a contradiction. Then, on a (dense) open subset Ω,there exists a non-zero U ∈ D , and a function µ ∈ C ∞ (Ω) such that Aξ = εµξ + U. (10)We discuss the shape operator depending on the causal character of the vector field U . • Suppose first that U is timelike or spacelike. There exists a unit W ∈ D and a function λ ∈ C ∞ (Ω) such that U = ε W λW , ε W = g ( W, W ) = ±
1. Then, Aξ = εµξ + ε W λW . Since g ( AW, X ) = 0 for any X ∈ D , AW = εg ( AW, ξ ) ξ = εg ( W, Aξ ) ξ = εg ( W, εµξ + ε W λW ) ξ = ελξ . We may write A ≡ εµ ελ . . . ε W λ ...0 , Aξ = εµξ + ε W λW, AW = ελξ. (11)Remark that g ( ξ, Aξ ) = µ and g ( AW, ξ ) = λ . Suppose first that rank( A ) = 1 on an open set˜Ω ⊂ M . It follows that λ = 0 on ˜Ω and therefore Aξ = εµξ , AX = 0, for all X ⊥ ξ. However,when AX is of the form AX = aX + bη ( x ) ξ , a, b ∈ C ∞ ( M ), we know that M is in the list of9heorem 1, which requires that a is always nonzero. This does not hold when rank( A ) = 1.We conclude that there exists a dense open subset Ω = { q ∈ M : λ ( q ) = 0 } in M such thatrank( A | Ω ) = 2 and A is given as in (11). We will work on this Ω.Next, given q ∈ Ω, consider an integral curve of ξ , namely, α : ( − δ, δ ) → Ω, with α (0) = q , α ′ ( s ) = ξ α ( s ) , ∀ s ∈ ( − δ, δ ). Note that ˜ ∇ α ′ ( s ) α ′ ( s ) = 0 because along the curve α ,˜ ∇ α ′ ( s ) α ′ ( s ) = ( ˜ ∇ ξ ξ ) α ( s ) = ( ∇ ξ ξ + εg ( Aξ, ξ ) N ) α ( s ) = ( φAξ + εµN ) α ( s ) = ( ε W λφW + εµN ) α ( s ) . The previous vector does not vanish since rank( A ) = 0 assures that λ ( α ( s )) = 0. • Suppose now that U is lightlike and Aξ = εµξ + U . There exist orthonormal vectors E and E in D , and a function a ∈ C ∞ (Ω) such that g ( E , E ) = 1, g ( E , E ) = − g ( E , E ) = 0, and U = aE − aE . Since g ( φE , φE ) = g ( J E , J E ) = +1, then n ≥
3. As before, AE i = ερ i ξ , for some function ρ i , i = 1 ,
2. But then, ρ i = g ( AE i , ξ ) = g ( E i , Aξ ) = g ( E i , εµξ + aE − aE ) = a . Therefore we may find an orthonormal basis B = { ξ, E , E , . . . , E n − } , with g ( E i , E i ) = ±
1, for which the shape operator is of the form A ≡ εµ εa εa . . . a − a . (12)As a consequence, AU = aAE − aAE = a ( εaξ − εaξ ) = 0 and AφU = aA ( φE − φE ) = εag ( φE − φE , Aξ ) ξ = εag ( φE − φE , εµξ + aE − aE ) ξ = εa [ − ag ( φE , E ) − ag ( φE , E )] =0 . Let I be a real interval such that 0 ∈ I and let α : I → C P np be a regular unit speedcurve in C P np , either spacelike or timelike. Given s ∈ I , let Π s be the complex holomorphicnon-degenerate hyperplane of maximal dimension in C P np tangent at α ( s ), such that Π ⊥ s =span { α ′ ( s ) , J α ′ ( s ) } . By Lemma 1, there exists a totally geodesic submanifold L s in C P np suchthat T α ( s ) L s = Π s . We know that L s is isometric to C P n − t , with t = p , if Π ⊥ is positivedefinite or t = p −
1, if Π ⊥ is negative definite. In this way, we obtain a family of totallygeodesic submanifolds { L s } s ∈ I , with the property that T α ( s ) L s = ( C α ′ ( s )) ⊥ . Intuitively, theset ∪ s ∈ I L s is a ruled real hypersurface. At first sight, there are two major families of ruledreal hypersurfaces in C P np , according to α ′ being timelike or spacelike. Remark 1.
The examples of ruled real hypersurfaces constructed in the way described abovesatisfy that the maximal holomorphic distribution D is integrable, where D = (span { α ′ , J α ′ } ) ⊥ .From now on, let L n − t be a connected open subset of C P n − t . Definition 3.
A smooth map f : M := I × L n − t → C P np is called a ruled hypersurfaceparametrization (shortly RHS-parametrization) induced by the curve α in the indefinite com-plex projective space C P np , iff f satisfies the following conditions:1. The curve α is unit speed g ( α ′ ( s ) , α ′ ( s )) = ± ε . . There exists a point x ∈ L n − t such that α : I → C P np , α ( s ) = f ( s, x ) for any s ∈ I ;3. For every s ∈ I , the map f s : L n − t → C P np , q f ( s, q ) is a totally geodesic andholomorphic immersion with ( f s ) ∗ T x L n − t = (span { α ′ ( s ) , J α ′ ( s ) } ) ⊥ , where t = p if ε = 1 , or t = p − if ε = − .4. For every v ∈ T q L n − t , the vector field Z v ∈ X α ( C P np ) , s ( Z v ) s :=( df s ) q ( v ) , along α , is parallel in the bundle (span { α ′ , J α ′ } ) ⊥ , i. e. Z v ( s ) ∈ span { α ′ ( s ) , J α ′ ( s ) } ⊥ and ˜ ∇ α ′ ( s ) Z v ∈ span { α ′ ( s ) , J α ′ ( s ) } , where ˜ ∇ is Levi Civita connection in T C P np . – In otherwords, f puts L n − t along α without rotations. Proposition 1.
Given a unit speed curve α : I → C P np , take s ∈ I . Then, there exists one,and only one, RHS-parametrization f : I × L n − t → C P np such that:1. there exists x ∈ C P n − t with α ( s ) = f s ( x ) ∈ f s ( C P n − t ) , and ( f s ) ∗ T x C P n − t =span { α ′ ( s ) , J α ′ ( s ) } ⊥ ;2. f ( s, x ) = α ( s ) for any s ∈ I . Remark 2.
The curve α might not be a Frenet curve. We are just asking α ′ ( s ) = 0 and notlightlike for all s . The vector field ˜ ∇ α ′ α ′ along α does exist, and can be spacelike, lightlike,timelike, zero at some points, and even a mixture of cases. Proof.
We are assuming that ε := g ( α ′ , α ′ ) = ± W = span { α ′ ( s ) , J α ′ ( s ) } ⊥ . If ε = 1, we take t = p , and if ε = −
1, we take t = p − v ∈ W , we consider the following initial value problem:˜ ∇ α ′ ( s ) Z v ( s ) = − ε h Z v ( s ) , ˜ ∇ α ′ ( s ) α ′ ( s ) i α ′ ( s ) − ε h Z v ( s ) , J ˜ ∇ α ′ ( s ) α ′ ( s ) i J α ′ ( s ) , (13) Z v ( s ) = v. Compute its solution Z v : I → T C P np . By (13), ˜ ∇ α ′ ( s ) Z v ( s ) ∈ span { α ′ ( s ) , J α ′ ( s ) } . Thesolution for v = 0 is just Z ( s ) = 0. Next, by taking inner product with α ′ and J α ′ in (13),it is simple to check h ˜ ∇ α ′ Z v , α ′ i = −h Z v , ˜ ∇ α ′ α ′ i , h ˜ ∇ α ′ Z v , J α ′ i = −h Z v , J ˜ ∇ α ′ α ′ i . With these, we obtain immediately that dds h Z v ( s ) , α ′ ( s ) i = h ˜ ∇ α ′ ( s ) Z v ( s ) , α ′ ( s ) i + h Z v ( s ) , ˜ ∇ α ′ ( s ) α ′ ( s ) i = 0 . And at s = s , h Z v ( s ) , α ′ ( s ) i = h v, α ′ ( s ) i = 0. This shows that Z v ⊥ α ′ . Similarly, Z v ( s ) ∈ span { α ′ ( s ) , J α ′ ( s ) } ⊥ for any s ∈ I . It is important here that we can extend thisconstruction to Z : I × W → T C P np , Z ( s, v ) = Z v ( s ), obtaining a smooth map.By Lemma 1, there exists a (unique) totally geodesic, holomorphic embedding Φ : C P n − t → C P np such that for some x ∈ C P n − t , Φ( x ) = α ( s ) and Φ ∗ ( T x C P n − t ) = W . By re-stricting to a suitable open subset L n − t of C P n − t with x ∈ L n − t , we obtain an injectivemap r : L n − t → T x C P n − t such that (exp x ) − = r . We construct the injective mapΨ : L n − t → W , Ψ := ( d Φ) x ◦ r. By combining these maps, we define f : I × L n − t → C P np , f ( s, x ) = exp α ( s ) ( Z ( s, Ψ( x ))) , f ( s, x ) = exp α ( s ) ( Z ( s )) = α ( s ) for any s ∈ I , and for each q ∈ L n − t , f ( s , q ) = exp α ( s ) ( Z ( s , Ψ( q ))) = exp α ( s ) (Ψ( q ))= exp Φ( x ) (cid:0) ( d Φ) x ( r ( q )) (cid:1) = Φ(exp x ( r ( q ))) = Φ( q ) . For every s ∈ I , the image of Z ( s, · ) is a holomorphic linear subspace of maximal dimension,thus f ( s, · ) is a totally geodesic submanifold, isometric to an open subset of C P n − t . Remark 3.
The uniqueness in these previous results are up to an isometry of C P n − t . Indeed,it is well known that totally geodesic hyperplanes C P n − are invariant by some subgroupsof isometries of C P n . Similarly, totally geodesic hyperplanes C P n − t are invariant by somesubgroups of isometries of C P np . Corollary 1.
Let M be a ruled real hypersurface in C P np . Then, for each point q ∈ M thereexists a unique parametrization of M around q as in Definition 3.Proof. We just need to consider the integral curve α : ( − δ, δ ) → C P np such that α (0) = q and α ′ = ξ α . We recall Proposition 1 to construct the parametrization. Given a point and aholomorphic hyperplane, by Lemma 1, the totally geodesic submanifolds C P n − t are unique,so the RHS-parametrization is a parametrization of M around q . We introduce the following definition, inspired by [1].
Definition 4 (Totally real circle) . A totally real circle is a Frenet curve of order , withconstant curvature and such that the two Frenet vectors F , F (see Definition 1) span atotally real plane at each point of the curve. Theorem 2.
A minimal ruled real hypersurface M in C P np is generated by a unit curve α : ( − δ, δ ) → M which satisfies one of the following conditions:a) α is a spacelike or timelike geodesic;b) α is a totally real circle of a totally real, totally geodesic R P , H or S in C P np ;c) α is a totally real curve, but not a Frenet curve, determined by the system F := α ′ , F := ˜ ∇ F F , ˜ ∇ F F = 0 , F lightlike, g ( F , F ) = 0 , g ( F , J F ) = 0 , (14) contained in a totally real, totally geodesic submanifold of C P np , namely B if α isspacelike, or B if α is timelike.Proof. As in Section 3.1, Aξ = εµξ + U , for some U ∈ D . Since g ( AX, Y ) = 0 for any
X, Y ∈ D , it follows that for a minimal ruled real hypersurface we have Aξ = U, AX = εg ( AX, ξ ) ξ = εg ( X, U ) ξ, φAX = 0 , (15)for any X ∈ D . We also need˜ ∇ ξ ξ = ∇ ξ ξ + εg ( Aξ, ξ ) N = φAξ = φU.
12e will discuss two cases, according to the causal character of the vector field U . Case 1 : Assume U is spacelike or timelike. We have˜ ∇ ξ ˜ ∇ ξ ξ = ˜ ∇ ξ φU = ∇ ξ φU + εg ( ξ, AφU ) N = ∇ ξ φU = ( ∇ ξ φ ) U + φ ∇ ξ U (6) = εg ( U, ξ ) Aξ − εg ( Aξ, U ) ξ + φ ∇ ξ U = − εg ( U, U ) ξ + φ ∇ ξ U. We may show that ∇ ξ U = 0. Its component in the direction of ξ is given by g ( ∇ ξ U, ξ ) = − g ( U, ∇ ξ ξ ) = − g ( U, φAξ ) = − g ( U, φU ) = 0 , while the one in the direction of U may be determined in the following way. We use theCodazzi equation (9) to evaluate ( ∇ ξ A ) U − ( ∇ U A ) ξ = εφU, (16)while using the definition for the covariant derivative of A gives( ∇ ξ A ) U − ( ∇ U A ) ξ = ∇ ξ AU − A ∇ ξ U − ∇ U Aξ + A ∇ U ξ = ∇ ξ ( εg ( U, U ) ξ ) − εg ( ∇ ξ U, U ) ξ − ∇ U U + AφAU =2 εg ( ∇ ξ U, U ) ξ + εg ( U, U ) φAξ − εg ( ∇ ξ U, U ) ξ − ∇ U U. We multiply by ξ in the above two equations and obtain0 = εg ( φU, ξ ) = εg ( ∇ ξ U, U ) g ( ξ, ξ ) − g ( ∇ U U, ξ )= g ( ∇ ξ U, U ) + g ( U, φAU ) = g ( ∇ ξ U, U ) . Let us show now that g ( ∇ ξ U, X ) = 0, for any X ⊥ U, ξ which is tangent to the hypersurface.As before, we evaluate in two ways( ∇ X A ) ξ − ( ∇ ξ A ) X (9) = − εφX = ∇ X Aξ − A ∇ X ξ − ∇ ξ AX + A ∇ ξ X = ∇ X U − AφAX + A ∇ ξ X, = ∇ X U + A ∇ ξ X, where we have used that AX = 0 for X ⊥ U . We take the component in the direction of ξ inthe above relation and obtain0 = − εg ( φX, ξ ) = g ( ∇ X U, ξ ) + g ( A ∇ ξ X, ξ )= − g ( U, ∇ X ξ ) + g ( ∇ ξ X, U ) = − g ( U, φAX ) + g ( ∇ ξ X, U )= − g ( X, ∇ ξ U ) . Eventually, it follows that ∇ ξ U = 0 and therefore˜ ∇ ξ ˜ ∇ ξ ξ = − εg ( U, U ) ξ. (17)Next, we consider α : ( − δ, δ ) → M the integral curve of ξ such that ξ α ( s ) = α ′ ( s ). We denoteby U s := U | α ( s ) and we may write˜ ∇ α ′ ( s ) ˜ ∇ α ′ ( s ) α ′ ( s ) = − εg ( U s , U s ) α ′ . (18)13 ase 1a) Suppose first that α is a geodesic. It follows that ˜ ∇ α ′ α ′ = 0 and therefore g ( U s , U s ) = 0. Since U s is not lightlike, it must hold that U s = 0. This implies ∇ ξ ξ | α ( s ) = φU s = 0 . (19) Case 1b)
Suppose that α is not a geodesic. Along the curve α we have˜ ∇ α ′ α ′ = φU s = 0 (20)and since U is not lightlike, we can normalize φU s , denoting: F ( s ) := 1 p | g ( U s , U s ) | φU s ,ε := sign( g ( U s , U s )) , ε κ = g ( U s , U s ) , where κ >
0. For F := α ′ and F defined as above, we have˜ ∇ F F = ε κ F . (21)We define F as F = ˜ ∇ F F + ε κ F , (22)where g ( F , F ) = ε and F could be a lightlike vector (see [16]). In fact, from (18) and (21)we obtain that ε κ ′ F − ε εκ F + ε κ F = − εg ( U s , U s ) F . (23)We need to check that { F , F , F } are orthogonal. If we multiply by F in (21), ε κ g ( F , F ) = g ( ˜ ∇ F F , F ) = 12 dds g ( F , F ) = 0 , g ( F , F ) . If we multiply with F in (22), we obtain g ( ˜ ∇ F F , F ) = − ε κ g ( F , F ) + g ( F , F ) ⇔− g ( F , ˜ ∇ F F ) = − κ + g ( F , F ) ⇔− g ( F , ε κ F ) = − κ + g ( F , F ) ⇔ g ( F , F ) . Similarly, it follows that g ( F , F ) = 0 when taking the inner product with F in (22). Fromthis fact, and as ε κ = g ( U s , U s ), κ = 0, we obtain that κ ′ = 0 and F = 0. This showsthat α is a curve of order 2 and κ is constant. In other words, α is a circle. Moreover, wewant to compute its holomorphic torsion, defined in [1], τ = g ( F , φF ). It follows directlyfrom (20) and (21): τ = g (cid:18) ξ α ( s ) , ε κ φ U s (cid:19) = 1 ε κ g ( ξ α ( s ) , − U s + εη ( U s ) ξ α ( s ) ) = 0 . Therefore, we conclude that α is a totally real circle.14oreover, we will prove that any totally real circle in C P np is actually contained in atotally geodesic and totally real surface Σ ⊂ C P np , as follows. Let α : I → C P np denote an arclength, totally real circle in C P np . Let t ∈ I , p = α ( t ) and F ( t ) , F ( t ) ∈ T α ( t o ) C P np , whichspan a totally real plane. Since α is a Frenet curve, then F and F are never lightlike. ByLemma 2, there exists a totally real and totally geodesic surface Σ in C P np such that p ∈ Σand T p Σ = span { F ( t ) , F ( t ) } . By Definition 4,˜ ∇ α ′ ( t ) α ′ ( t ) = ε κ E , ˜ ∇ α ′ ( t ) E = − ε κ E , (24)where ˜ ∇ is the Levi-Civita connection on C P np and α ′ ( t ) = E . Consider now a curve β : I → Σ contained in the surface Σ in C P np , such that p = β ( t ) and β satisfies the followingFrenet system of equations in Σ: ∇ β ′ β ′ = ε κ E , ∇ β ′ E = − ε κ E , where ∇ is the Levi-Civita connection on Σ. We impose, additionally, that α and β satisfy thesame initial conditions, α ( t ) = β ( t ), β ′ ( t ) = F ( t ) = E ( t ) and E ( t ) = F ( t ). Since Σis totally geodesic in C P np , i.e. ˜ ∇ X Y = ∇ X Y for any vectors X, Y tangent to Σ ⊂ C P np , wehave that the Frenet equations of β are the same both in Σ and in C P np . Therefore, β alsosatisfies (24). By recalling the uniqueness of solution to initial value problems, then α = β .That is, α is contained in the totally geodesic, totally real surface Σ. The type of surface willdepend on the causal character of the Frenet system { F = α ′ , F } of α . To explain this, weresort to Lemma 2. Take Π = span { F , F } at a given point.1. If Π is spacelike, then the surface is (an open subset of) R P .2. If Π is an indefinite plane, the surface is locally isometric to S .3. If Π is negative definite, the surface is H . Case 2.
Assume U is lightlike. Again, we have˜ ∇ ξ ξ = ∇ ξ ξ + εg ( Aξ, ξ ) N = φAξ = φU, (25)from which ˜ ∇ ξ ˜ ∇ ξ ξ = ˜ ∇ ξ φU = ∇ ξ φU + εg ( U, φU ) N = ∇ ξ φU = ( ∇ ξ φ ) U + φ ∇ ξ U (6) = εg ( U, ξ ) Aξ − εg ( Aξ, U ) ξ + φ ∇ ξ U = φ ∇ ξ U. (26)Further on, we show that ∇ ξ U = 0. Indeed, firstly we have g ( ∇ ξ U, ξ ) = − g ( U, ∇ ξ ξ ) = − g ( U, φU ) = 0 ,g ( ∇ ξ U, U ) = 12 ξ ( g ( U, U )) = 0 . and we may choose X ∈ D , such that g ( X, U ) = ε . This gives immediately that AX = εg ( AX, ξ ) ξ = εg ( X, U ) ξ = ξ . Next, we compute in two different ways ( ∇ X A ) ξ − ( ∇ ξ A ) X :( ∇ X A ) ξ − ( ∇ ξ A ) X = − εφX = ∇ X Aξ − A ∇ X ξ − ( ∇ ξ AX − A ∇ ξ X )= ∇ X U − AφAX − ∇ ξ ξ + A ∇ ξ X.
15e take the components in the direction of ξ and obtain that0 = g ( ∇ X U, ξ ) + g ( ∇ ξ X, U ) = − g ( X, ∇ ξ U ) . We conclude therefore that ∇ ξ U = 0 and so, equation (26) implies that ˜ ∇ ξ ˜ ∇ ξ ξ = 0.As before, we consider the integral curve α of ξ , that is α : ( − δ, δ ) → M and α ′ ( s ) = ξ | α ( s ) ,for the arc parameter s . We will use the index “ s ” for vector fields along the curve α . By(25), we obtain that α is determined by the following system of equations: F = α ′ = ξ s , ˜ ∇ F F = φU s =: F , ˜ ∇ F F = 0 , (27) g ( F , F ) = 0 , g ( F , J F ) = 0 , where F is a lightlike vector.Let us show that the curve α is contained in a 3-dimensional submanifold in C P np . Wehave that F = α ′ = ξ s , J F = N s and 0 = F ∈ span { F , J F } ⊥ = D . There exists a localorthonormal basis of D of the form { w , J w , . . . , w n − , J w n − } such that the first t vectors w , . . . , w t are timelike and the remaining vectors w t +1 , . . . , w n − are spacelike. There exist a i , b i functions for i = 1 , . . . , n −
1, such that we may write F = P n − k =1 ( a k w k + b k J w k ), Ifwe denote by v k := a k w k + b k J w k , this is equivalent to F = P k v k . Moreover, the timelikevector E = P tk =1 v k and the spacelike vector E = P n − k = t +1 v k satisfy F = E + E and g ( E , J E ) = 0. Given the totally real Q := span { F (0) , E , E } ⊂ D α (0) , there exists atotally geodesic, totally real, 3-dimensional submanifold B of C P np , for which α (0) ∈ B , F (0) ∈ Q and T α (0) B = Q . Given the uniqueness of solution for the Cauchy problem in (27),we have that the curve α is entirely contained in B .To discribe B , we resort to Lemma 3. To make it more simple and understandable, wework with the previous descriptions to Lemma 3. Case c-1) B = dS . Let ¯ D and D the Levi-Civita connections of L and dS , respectively.Bearing in mind that the position vector χ : dS → L is a unit spacelike normal vector field,the Gauss equation is ¯ D X Y = D X Y − h X, Y i χ, for any X, Y ∈ Γ( T dS ), where h , i is the standard flat metric in L .Let α : I → dS be a unit, spacelike curve in the de Sitter 3-space, with F = α ′ . Bycombining the Gauss equation and (14), we obtain the following ODE, F = D F F = ¯ D F F + h α ′ , α ′ i α = α ′′ + α, D F F = α ′′′ + α ′ . As a consequence, the vector F along α must be constant, lightlike, and orthogonal to α ′ .Thus, everything reduces to the following initial value problem: α ′′ + α = F , α (0) = p , α ′ (0) = v , p , v , F ∈ L , with { p , v , F } orthogonal, { p , v } unit spacelike, and F lightlike. The solution is α : R → dS , α ( s ) = F + cos( s )( p − F ) + sin( s ) v . Indeed, h α ( s ) , α ( s ) i = 1, h α ′ ( s ) , α ′ ( s ) i = 1, and α ′′ ( s ) + α ( s ) = F .Suppose that α : I → dS could be unit, timelike. As before, at a certain point, wewould reach to a set of vectors p = α ( s ) unit spacelike, v = α ′ ( s ) unit timelike, p ⊥ v .16owever, if v is timelike, since dim dS = 3 and its index is 1, then F ∈ v ⊥ cannot belightlike. Case c-2) B = S . According to Lemma 3, we consider the hyperquadric ˆ B = { ( x , . . . , x ) ∈ R : − x − x + x + x = 1 } , which is locally anti-isometric to the anti-de Sitter 3-space. Let¯ D and D the Levi-Civita connections of R and ˆ B , respectively. Bearing in mind that theposition vector χ : ˆ B → R is a unit spacelike normal vector field, the Gauss equation is¯ D X Y = D X Y − h X, Y i χ, for any X, Y ∈ Γ( T ˆ B ), where h , i is the standard flat metric in R .Let α : I → ˆ B be a unit, timelike curve in ˆ B , with F = α ′ . By combining the Gaussequation and (14), we obtain the following ODE, F = D F F = ¯ D F F + h α ′ , α ′ i α = α ′′ − α, D F F = α ′′′ − α ′ . As a consequence, the vector F along α must be constant, lightlike, and orthogonal to α ′ .Thus, everything reduces to the following initial value problem: α ′′ − α = F , α (0) = p , α ′ (0) = v , p , v , F ∈ L , with orthogonal { p , v , F } , p unit spacelike, v unit timelike, and F lightlike. The solutionis α : R → ˆ B , α ( s ) = cosh( s )( p + F ) + sinh( s ) v − F . Suppose that α : I → ˆ B could be unit, spacelike. As before, at a certain point, we wouldreach to a set of vectors p = α ( s ) unit spacelike, v = α ′ ( s ) unit spacelike. However, sincedim ˆ B and the index is 2, if v is spacelike, then F ∈ v ⊥ cannot be lightlike. We develop the first example in detail, and simplify the rest due to the similarities.
Example 1.
We take 1 ≤ p ≤ n − n ≥
3. Consider Ω = { z = ( z , . . . , z n ) ∈ S n − p : z n = 0 } .We define the mapˆ ψ : ˆ M = R × Ω → S n +12 p , ˆ ψ ( t, z ) = ( z , . . . , z n − , cos( t ) z n , sin( t ) z n ) . Since p ≤ n −
1, the matrix A t = I n − t ) − sin( t )0 sin( t ) cos( t ) ∈ SU ( p, n + 1 − p ) , and so, it represents an isometry of S n +12 p , which also induces an isometry of C P np . Define foreach t ∈ R , Ω t := ˆ ψ ( { t } × Ω). Thus, A t ( ˆ ψ ( { } × Ω)) = A t (Ω ). Each Ω t , t ∈ R , is totallygeodesic, and so P i (Ω t ) ⊂ C P np is an open subset of a totally geodesic complex hyperplane.With them, we construct the following ruled real hypersuface in C P np , ψ : M = R × P i (Ω) → C P np , ψ ( t, [ z ]) = P i ( ˆ ψ ( t, z )) , ψ ( M ) = [ t ∈ R P i (Ω t ) . ψ isˆ ψ t = ∂ t ˆ ψ = (0 , . . . , , − sin( t ) z n , cos( t ) z n ) ,d ˆ ψ (0 , X ) = ( X , . . . , X n − , cos( t ) X n , sin( t ) X n ) , X ∈ T z Ω . If we recall iz = J χ ( z ), the vertical part of P i : S n +12 p → C P np is V z = Span { iz } =Span { J χ ( z ) } . Then, J χ | ˆ ψ ( t,z ) = i ˆ ψ ( t, z ) = d ˆ ψ ( t,z ) (0 , iz ) . Given ( t, z ) ∈ R × Ω, and X ∈ T z Ω, then h z, X i = 0. Simple computations lead to h i ˆ ψ, d ˆ ψ (0 , X ) i = h iz, X i , h ∂ t ˆ ψ, d ˆ ψ (0 , X ) i = 0 , h d ˆ ψ (0 , X ) , d ˆ ψ (0 , Y ) i = h X, Y i , h ∂ t ˆ ψ, ∂ t ˆ ψ i = | z n | > . According to this, we defineˆ ξ ˆ ψ ( t,z ) = ∂ t ˆ ψ | z n | , ˆ N ˆ ψ ( t,z ) = i ˆ ξ ˆ ψ ( t,z ) = 1 | z n | (0 , . . . , , − i sin( t ) z n , i cos( t ) z n ) . Let us see that ˆ N is a horizontal, spacelike, unit normal vector field along ˆ ψ : h ˆ N , ∂ t ˆ ψ i = 1 | z n | h i∂ t ˆ ψ, ∂ t ˆ ψ i = 0 , h ˆ N , ˆ N i = 1 | z n | h i∂ t ˆ ψ, i∂ t ˆ ψ i = 1 , h ˆ N , d ˆ ψ (0 , X ) i = 1 | z n | Re (cid:0) − sin( t ) iz n cos( t ) X n + cos( t ) iz n sin( t ) X n (cid:1) = 0 , h ˆ N , i ˆ ψ i = h i ˆ ξ, i ˆ ψ i = h ˆ ξ, ˆ ψ i = 1 | z n | h ∂ t ˆ ψ, ˆ ψ i = 0 . We point out that an integral curve of ˆ ξ is α ( s ) = ˆ ψ ( t + s/ | z n | , z ) = ( z , . . . , z n − , cos ( t + s/ | z n | ) z n , sin ( t + s/ | z n | ) z n ) . (28)Indeed, α (0) = ˆ ψ ( t, z ), α ′ (0) = dds ˆ ψ ( t + s/ | z n | , z ) (cid:12)(cid:12)(cid:12) s =0 = | z n | ∂ t ˆ ψ ( t, z ) = ˆ ξ ˆ ψ ( t,z ) , and α ′ ( s ) = 1 | z n | (0 , . . . , , − sin ( t + s/ | z n | ) z n , cos ( t + s/ | z n | ) z n ) = ˆ ξ ˆ ψ ( t + s/ | z n | ,z ) = ˆ ξ α ( s ) . Let ˆ A be the shape operator associated with ˆ N . By (5), we obtainˆ A ˆ ξ = − ˆ ∇ ˆ ξ ˆ N = − D ˆ ξ ˆ N = − dds ˆ N α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds ˆ N ˆ ψ ( t + s/ | z n | ,z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds i | z n | (0 , . . . , , − sin ( t + s/ | z n | ) z n , cos ( t + s/ | z n | ) z n ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 1 | z n | (0 , . . . , , i cos( t ) z n , i sin( t ) z n ) . Finally, h ˆ A ˆ ξ, ˆ ξ i = 1 | z n | h (0 , i cos( t ) z n , i sin( t ) z n ) , (0 , − sin( t ) z n , cos( t ) z n ) i = 1 | z n | Re (cid:0) − i cos( t ) sin( t ) | z n | + i sin( t ) cos( t ) | z n | (cid:1) = 0 . M is N = P i ∗ ( ˆ N ), we also have ξ = P i ∗ ( ˆ ξ ), andthe shape operator A of M is A = P i ∗ ˆ A , as expected. With all this information, we obtain g ( Aξ, ξ ) = 0. As g ( AX, Y ) = 0 for any
X, Y ⊥ ξ , X, Y ∈ T M , then M is minimal.Let us study now the integral curves of ˆ ξ . Note that h α ′ , iα i = 0, which means that α isa horizontal curve. By (5), F ( s ) := ˆ ∇ α ′ ( s ) α ′ ( s ) = D α ′ ( s ) α ′ ( s ) + h α ′ ( s ) , α ′ ( s ) i α ( s ) = α ′′ ( s ) + α ( s )= 1 | z n | (0 , . . . , , − cos( t + s/ | z n | ) z n , − sin( t + | z n | ) z n ) + α ( s )= (cid:18) z , . . . , z n − , (cid:18) − | z n | (cid:19) cos( t + s/ | z n | ) z n , (cid:18) − | z n | (cid:19) sin( t + s/ | z n | ) z n (cid:19) ; h F ( s ) , F ( s ) i = − p X j =1 | z j | + n − X j = p +1 | z j | + (cid:18) − | z n | (cid:19) | z n | = 1 + "(cid:18) − | z n | (cid:19) − | z n | = 1 | z n | − . When | z n | = 1, F ( s ) is zero or lightlike. The first case holds when z = (0 , . . . , , e ir ), and thesecond one, otherwise. When | z n | < F ( s ) is always spacelike. If | z n | > F ( s ) is alwaystimelike. In addition, similar computations give h F ( s ) , α ( s ) i = h F ( s ) , iα ( s ) i = 0, which showthat F ( s ) is a horizontal vector along α . We need this to project it safely to C P np . • Case | z n | = 1. Then, F ( s ) = ( z , . . . , z n − , , P i ( α ( s )) is the case a ) or c ) of Theorem 2. • Case | z n | <
1. Then, F ( s ) is spacelike. To compute the Frenet system, we note that1 − / | z n | does not depend on s . We obtain κ := s | z n | − > , F ( s ) = α ′ ( s ) ,F ( s ) := F ( s ) κ = (cid:18) z κ , . . . , z n − κ , − κ cos( s/ | z n | ) z n , − κ ( s ) sin( s/ | z n | ) z n (cid:19) , ˆ ∇ α ′ ( s ) α ′ ( s ) = κ F ( s ) . Next, since h F , α ′ i = 0, by (5),ˆ ∇ α ′ ( s ) F ( s ) = F ′ ( s ) − h α ′ ( s ) , F ( s ) i α ( s ) = − κ α ′ ( s ) . This is a Frenet curve of order 2, with F , F spacelike, with constant curvature κ and h J F ( s ) , F ( s ) i = 1 | z n | h (0 , . . . , , − i sin( s/ | z n | ) z n , i cos( s/ | z n | ) z n ) , F ( s ) i = 0 . Thus, if we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2. • Case | z n | >
1. Then, F ( s ) is timelike. Similar computations give κ := s − | z n | > , F ( s ) = α ′ ( s ) ,F ( s ) := − F ( s ) κ = (cid:18) − z κ , . . . , − z n − κ , − κ cos( s/ | z n | ) z n , − κ sin( s/ | z n | ) z n (cid:19) , ˆ ∇ α ′ ( s ) α ′ ( s ) = − κ F ( s ) .
19e also obtain h J F ( s ) , F ( s ) i = 0. Next, since h F , α ′ i = 0, by (5),ˆ ∇ α ′ ( s ) F ( s ) = F ′ ( s ) − h α ′ ( s ) , F ( s ) i α ( s ) = − κ α ′ ( s ) . If we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2.We define the function f : (0 , π/ → R , f ( r ) = √ r ). Clearly, for 0 < r < π/ < f ( r ) < f ( π/
4) = 1; and for π/ < r ≤ π/ f ( r ) >
1. With this, we consider the curve γ : (0 , π/ → Ω ⊂ S n − p , γ ( r ) = (1 , , . . . , , √ r ) , √ r )). In particular, the points γ ( π/ γ ( π/
4) and γ ( π/
2) lie in the same connected component of Ω, satifying | z n | < | z n | = 1 and | z n | = √
2, respectively. Thus, there exist different points lying in the sameconnected component of ˆ M , such that the integral curves of ˆ ξ passing through them behavein a very different way.By projecting everything to C P np , we have a minimal ruled real hypersurface in C P np suchthat certain integral curves of ξ , starting at different points at the same connected component,behave as in any case of Theorem 2. Example 2.
We take 1 ≤ p ≤ n − n ≥
4. Consider Ω = { z = ( z , . . . , z n ) ∈ S n − p : z = 0 } .We define the mapˆ ψ : ˆ M := R × Ω → S n +12 p , ˆ ψ ( t, z ) = (cosh( t ) z , z . . . , z n , sinh( t ) z ) . Since p ≤ n −
1, the matrix A t = cosh( t ) 0 sinh( t )0 I n − t ) 0 cosh( t ) ∈ SU ( p, n + 1 − p ) , and it represents an isometry of S n +12 p , inducing an isometry of C P np . Define for each t ∈ R ,Ω t := ˆ ψ ( { t } × Ω) = A t ( ˆ ψ ( { } × Ω)) = A t (Ω ). Each Ω t , t ∈ R , is totally geodesic, and so P i (Ω t ) ⊂ C P np is a totally geodesic complex hyperplane. Then, we construct the followingruled real hypersuface in C P np , ψ : M = R × P i (Ω) → C P np , ψ ( t, [ z ]) = P i ( ˆ ψ ( t, z )) , ψ ( M ) = [ t ∈ R P i (Ω t ) . The differential of ˆ ψ is ∂ t ˆ ψ = (sinh( t ) z , , . . . , , cosh( t ) z ) ,d ˆ ψ (0 , X ) = (cosh( t ) X , X . . . , X n , sinh( t ) X ) , X ∈ T z Ω . Similarly to the previous example, h i ˆ ψ, d ˆ ψ (0 , X ) i = h iz, X i , h ∂ t ˆ ψ, d ˆ ψ (0 , X ) i = 0 , h d ˆ ψ (0 , X ) , d ˆ ψ (0 , Y ) i = h X, Y i , h ∂ t ˆ ψ, ∂ t ˆ ψ i = | z | > , ˆ ξ ˆ ψ ( t,z ) = ∂ t ψ | z | , ˆ N ˆ ψ ( t,z ) = i ˆ ξ ˆ ψ ( t,z ) = i | z | (sinh( t ) z , , . . . , , cosh( t ) z ) . Now, ˆ N is a horizontal, spacelike, unit normal vector field along ˆ ψ . An integral curve of ˆ ξ is α ( s ) = ˆ ψ ( t + s/ | z | , z ) = (cosh ( t + s/ | z | ) z , z , . . . , z n , sinh ( t + s/ | z | ) z ) . (29)20et ˆ A be the shape operator associated with ˆ N . Then, by (5),ˆ A ˆ ξ = − ˆ ∇ ˆ ξ ˆ N = − D ˆ ξ ˆ N = − dds ˆ N α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds ˆ N ˆ ψ ( t + s/ | z | ,z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds i | z | (sinh ( t + s/ | z | ) z , , . . . , , cosh ( t + s/ | z | ) z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − i | z | (cosh( t ) z , , . . . , , sinh( t ) z ) ; h ˆ A ˆ ξ, ˆ ξ i = 0 . Again, ψ ( M ) is a minimal ruled real hypersurface in C P np .Let us study now the integral curves of ˆ ξ . Note that h α ′ , iα i = 0, which means that α isa horizontal curve. By (5), F ( s ) := ˆ ∇ α ′ ( s ) α ′ ( s ) = D α ′ ( s ) α ′ ( s ) + h α ′ ( s ) , α ′ ( s ) i α ( s ) = α ′′ ( s ) + α ( s )= 1 | z | (cosh ( t + s/ | z | ) , , . . . , , sinh( t + s/ | z | ) z ) + α ( s )= (cid:18)(cid:18) | z | (cid:19) cosh( t + s/ | z | ) z , z , . . . , z n , (cid:18) | z | (cid:19) sinh( t + s/ | z | ) z (cid:19) ; h F ( s ) , F ( s ) i = − . By now, the Frenet system is κ := s | z | + 1 > , F ( s ) = α ′ ( s ) ,F ( s ) := − F ( s ) κ = (cid:18) − κ cosh( t + s/ | z | ) z , − z κ , . . . , − z n κ , − κ sinh( t + s/ | z | ) z (cid:19) ,ε = h α ′ , α ′ i = +1 , ε = h F , F i = − , ˆ ∇ F ( s ) F ( s ) = − κ F ( s ) = ε κ F ( s ) . We also obtain h J F ( s ) , F ( s ) i = 0. Next, since h F , α ′ i = 0, by (5),ˆ ∇ F ( s ) F ( s ) = F ′ ( s ) + h F ( s ) , F ( s ) i α ( s ) = − κ F ( s ) . If we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2. Example 3.
We take 2 ≤ p ≤ n − n ≥
3, and Ω = { z = ( z , . . . , z n ) ∈ S n − p − : z n = 0 } .We define the mapˆ ψ : ˆ M = R × Ω → S n +12 p , ˆ ψ ( t, z ) = (sinh( t ) z n , z , . . . , z n − , cosh( t ) z n ) . We use A t as in Example 2. Define for each t ∈ R , Ω t := ˆ ψ ( { t } × Ω) = A t ( ˆ ψ ( { } × Ω)) = A t (Ω ). Again, Ω t is also totally geodesic, and so P i (Ω t ) ⊂ C P np is a totally geodesic complexhyperplane. Then, we construct the following ruled real hypersurface ψ : M = R × P i (Ω) → C P np , ψ ( t, [ z ]) = P i ( ˆ ψ ( t, z )) , ψ ( M ) = [ t ∈ R P i (Ω t ) . ψ is ∂ t ˆ ψ = (cosh( t ) z n , , . . . , , sinh( t ) z n ) ,d ˆ ψ (0 , X ) = (sinh( t ) X n , X . . . , X n − , cosh( t ) X n ) , X ∈ T z Ω . Similarly to previous examples, h i ˆ ψ, d ˆ ψ (0 , X ) i = h iz, X i , h ∂ t ˆ ψ, d ˆ ψ (0 , X ) i = 0 , h d ˆ ψ (0 , X ) , d ˆ ψ (0 , Y ) i = h X, Y i , h ∂ t ˆ ψ, ∂ t ˆ ψ i = −| z n | < , ˆ ξ ˆ ψ ( t,z ) = ∂ t ψ | z n | , ˆ N ˆ ψ ( t,z ) = i ˆ ξ ˆ ψ ( t,z ) = i | z n | (cosh( t ) z n , , . . . , , sinh( t ) z n ) . Now, ˆ N is a horizontal, timelike, unit normal vector field along ˆ ψ . An integral curve of ˆ ξ is α ( s ) = ˆ ψ ( t + s/ | z n | , z ) = (sinh ( t + s/ | z n | ) z n , z , . . . , z n − , cosh ( t + s/ | z n | ) z n ) . (30)Let ˆ A be the shape operator associated with ˆ N . Then, by (5),ˆ A ˆ ξ = − ˆ ∇ ˆ ξ ˆ N = − D ˆ ξ ˆ N = − dds ˆ N α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds ˆ N ˆ ψ ( t + s/ | z n | ,z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds i | z n | (cosh ( t + s/ | z n | ) z n , , . . . , , sinh ( t + s/ | z n | ) z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − i | z n | (sinh( t ) z n , , . . . , , cosh( t ) z n ) ; h ˆ A ˆ ξ, ˆ ξ i = 0 . As before, ψ ( M ) is a minimal ruled real hypersurface in C P np .Let us study now the integral curves of ˆ ξ . Note that h α ′ , iα i = 0, which means that α isa horizontal curve. By (5), F ( s ) := ˆ ∇ α ′ ( s ) α ′ ( s ) = D α ′ ( s ) α ′ ( s ) + h α ′ ( s ) , α ′ ( s ) i α ( s ) = α ′′ ( s ) − α ( s )= 1 | z n | (sinh ( t + s/ | z n | ) , , . . . , , cosh( t + s/ | z n | ) z n ) − α ( s )= (cid:18)(cid:18) | z n | − (cid:19) sinh( t + s/ | z n | ) z n , − z , . . . , − z n − , (cid:18) | z n | − (cid:19) cosh( t + s/ | z n | ) z n (cid:19) ; h F ( s ) , F ( s ) i = 1 | z n | − . When | z n | = 1, F ( s ) is zero or lightlike. The first case holds when z = (0 , . . . , , e ir ), andthe second one otherwise. When | z n | < F ( s ) is spacelike. And when | z n | > F ( s ) istimelike. In addition, similar computations give h F ( s ) , α ( s ) i = h F ( s ) , iα ( s ) i = 0, which showthat F ( s ) is a horizontal vector along α . We need this to project it safely to C P np . • Case | z n | = 1. Then, F ( s ) = (0 , z , . . . , z n − , P i ( α ( s )) is the case a ) or c ) of Theorem 2.22 Case | z n | <
1. Then, F ( s ) is spacelike. We compute the Frenet system. κ := s | z n | − > , F ( s ) = α ′ ( s ) , ε = − ,F ( s ) := F ( s ) κ = (cid:18) κ sinh( t + s/ | z n | ) z n , − z κ , . . . , − z n − κ , κ ( s ) cosh( t + s/ | z n | ) z n (cid:19) ,ε = h F ( s ) , F ( s ) i = 1; ˆ ∇ F ( s ) F ( s ) = κ F ( s ) = ε κ F ( s ) . Next, as h F , α ′ i = 0, by (5),ˆ ∇ F ( s ) F ( s ) = F ′ ( s ) + h F ( s ) , F ( s ) i α ( s ) = κ F ( s ) = − ε κ F ( s ) . This is a Frenet curve of order 2, with F timelike, F spacelike, constant curvature κ and h J F ( s ) , F ( s ) i = 1 | z n | h ( i cosh( t + s/ | z n | ) z n , , . . . , , i sinh( t + s/ | z n | ) z n ) , F ( s ) i = 0 . Thus, if we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2. • Case | z n | >
1. Then, F ( s ) is timelike. Similar computations give κ := s − | z n | > , F ( s ) = α ′ ( s ) , ε = − ,F ( s ) := − F ( s ) κ = (cid:18) − κ sinh( t + s/ | z n | ) z n , z κ , . . . , z n − κ , − κ cosh( t + s/ | z n | ) z n (cid:19) , h F ( s ) , F ( s ) i = −
1; ˆ ∇ F ( s ) F ( s ) = − κ F ( s ) = ε κ F ( s ) . We also obtain h J F ( s ) , F ( s ) i = 0. Next, as h F , α ′ i = 0, by (5),ˆ ∇ F ( s ) F ( s ) = F ′ ( s ) + h α ′ ( s ) , F ( s ) i α ( s ) = − κ F ( s ) = ε κ F ( s ) . This is a Frenet curve of order 2, with F , F timelike, constant curvature κ and totally real.If we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2.As in Example 1, there exist different points lying in the same connected component ofˆ M , such that the integral curves of ˆ ξ passing through them behave in a very different way.Projecting everything to C P np , we have a minimal ruled real hypersurface in C P np such thatcertain integral curves of ξ , starting at different points at the same connected component,behave as in any case of Theorem 2. Example 4.
We take 2 ≤ p ≤ n − n ≥
3. Consider Ω = { z = ( z , . . . , z n ) ∈ S n − p : z = 0 } .We define the mapˆ ψ : R × Ω → S n +12 p , ˆ ψ ( t, z ) = (sin( t ) z , cos( t ) z , z . . . , z n ) . We use the matrix A t as in Example 1. Define for each t ∈ R , Ω t := ˆ ψ ( { t } × Ω) = Ω t = A t ( ˆ ψ ( { } × Ω)) = A t (Ω ). Again, Ω t is also totally geodesic, and so P i (Ω t ) ⊂ C P np is atotally geodesic complex hyperplane. Then, we have the following ruled real hypersurface ψ : M = R × P i (Ω) → C P np , ψ ( t, [ z ]) = P i ( ˆ ψ ( t, z )) , ψ ( M ) = [ t ∈ R P i (Ω t ) . ∂ t ˆ ψ = (cos( t ) z , − sin( t ) z , , . . . , ,d ˆ ψ (0 , X ) = (sin( t ) X , cos( t ) X , X , . . . , X n ) , X ∈ T z Ω . h i ˆ ψ, d ˆ ψ (0 , X ) i = h iz, X i , h ∂ t ˆ ψ, d ˆ ψ (0 , X ) i = 0 , h d ˆ ψ (0 , X ) , d ˆ ψ (0 , Y ) i = h X, Y i , h ∂ t ˆ ψ, ∂ t ˆ ψ i = −| z | < . ˆ ξ ˆ ψ ( t,z ) = ∂ t ˆ ψ | z | , ˆ N ˆ ψ ( t,z ) = i ˆ ξ ˆ ψ ( t,z ) = 1 | z | ( i cos( t ) z , − i sin( t ) z , , . . . , . In addition, ˆ N is a horizontal, timelike, unit normal vector field along ˆ ψ . We point out thatan integral curve of ˆ ξ is α ( s ) = ˆ ψ ( t + s/ | z | , z ) = (sin ( t + s/ | z | ) z , cos ( t + s/ | z | ) z , z . . . , z n ) . (31)Indeed, α (0) = ˆ ψ ( t, z ), α ′ (0) = dds ˆ ψ ( t + s/ | z | , z ) (cid:12)(cid:12)(cid:12) s =0 = | z | ∂ t ˆ ψ ( t, z ) = ˆ ξ ˆ ψ ( t,z ) , and α ′ ( s ) = 1 | z | (cos ( t + s/ | z | ) z , − sin ( t + s/ | z | ) z , , . . . ,
0) = ˆ ξ ˆ ψ ( t + s/ | z | ,z ) = ˆ ξ α ( s ) . Let ˆ A be the shape operator associated with ˆ N . Then, by (5),ˆ A ˆ ξ = − ˆ ∇ ˆ ξ ˆ N = − D ˆ ξ ˆ N = − dds ˆ N α ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds ˆ N ˆ ψ ( t + s/ | z | ,z ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = − dds i | z | (cos ( t + s/ | z | ) z , − sin ( t + s/ | z | ) z , , . . . , (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 1 | z | ( − i sin( t ) z , i cos( t ) z , , . . . ,
0) ; h ˆ A ˆ ξ, ˆ ξ i = 0 . As before, ψ ( M ) is a minimal ruled real hypersurface in C P np .Let us study now the integral curves of ˆ ξ . By (5), F ( s ) := ˆ ∇ α ′ ( s ) α ′ ( s ) = D α ′ ( s ) α ′ ( s ) + h α ′ ( s ) , α ′ ( s ) i α ( s ) = α ′′ ( s ) − α ( s )= − (cid:18)(cid:18) | z | + 1 (cid:19) sin( t + s/ | z | ) z , (cid:18) | z | + 1 (cid:19) cos( t + s/ | z | ) z , z , . . . , z n (cid:19) , h F ( s ) , F ( s ) i = − . Given our computations so far, the Frenet system is κ := s | z | + 1 > , F ( s ) = α ′ ( s ) ,F ( s ) := − F ( s ) κ = (cid:18) − κ sin( t + s/ | z | ) z , − κ cos( t + s/ | z | ) z , − z κ , . . . , − z n κ (cid:19) ,ε = h α ′ , α ′ i = +1 , ε = h F , F i = − , ˆ ∇ F ( s ) F ( s ) = − κ F ( s ) = ε κ F ( s ) . We also obtain h J F ( s ) , F ( s ) i = 0. Next, since h F , α ′ i = 0, by (5),ˆ ∇ F ( s ) F ( s ) = F ′ ( s ) + h F ( s ) , F ( s ) i α ( s ) = F ′ ( s ) = − κ F ( s ) . If we project to C P np , P i ( α ) is a curve of type b ) in Theorem 2.24 eferences [1] T. Adachi, S. Maeda, S. Udagawa, Circles in a complex projective space , Osaka J. Math.32 (1995), 709–719.[2] H. Anciaux and K. Panagiotidou, Hopf hypersurfaces in pseudo-Riemennian complexand para-complex space forms,
Diff. Geom. Appl. (2015), 1-14.[3] M. Barros, A. Romero, Indefinite K¨ahler Manifolds , Math. Ann. (1982), 55-62.[4] D.E. Blair, Riemannian Geometry of contact and symplectic manifolds,
Progress in Math-ematics (2002), Birkhauser Boston Inc. Boston.[5] T.E. Cecil and P.J. Ryan,
Focal sets and real hypersurfaces in complex projective space ,Trans. Amer. Math. Soc. (1982), 481-499.[6] T.E. Cecil and P.J. Ryan, Geometry of hypersurfaces, Springer Monographs in Mathe-matics (2015), Springer Science, New York.[7] D. Gromoll, W.Klingenberg, W. Meyer:
Riemannsche Geometrie im Großen , Springer,Berlin, 1968.[8] M. Kimura,
Real hypersurfaces and complex submanifolds in complex projective space , Trans. A.M.S. (1986), 137-149.[9] M. Kimura,
Sectional curvatures of holomorphic planes of a real hypersurface in P n ( C ), Math. Ann. (1987), 487-497.[10] M. Kimura, S. Maeda, H. Tanabe,
New construction of ruled real hypersurfaces in acomplex hyperbolic space and its applications , Geometriae Dedicata (2020) 207:227–242.https://doi.org/10.1007/s10711-019-00496-4[11] M. Kimura, M. Ortega,
Hopf Real Hypersurfaces in Indefinite Complex Projective Space ,Mediterr. J. Math. (2019) 16:27. https://doi.org/10.1007/s00009-019-1299-9.[12] M. Lohnherr and H. Reckziegel, On ruled real hypersurfaces in complex space forms,
Geom. Dedicata (1999), 267-286.[13] S. Montiel, Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. (3),515-535 (1985).[14] B. O’Neill, The fundamental equations of a submersion , Michigan Math. J. 13 (1966),459–469.[15] B. O’Neill,
Semi-Riemannian Geometry. With Applications to Relativity , Pure and Ap-plied Mathematics, 103. Academic Press, Inc., New York, 1983.[16] A. G. Pastor,
Geometr´ıa de curvas degeneradas , PhD thesis – Universidad de Murcia,Departamento de Matemat´ıcas, 2002, advisors P. L. Saor´ın, J. L. Garc´ıa Hern´andez.[17] O. P´erez-Barral,
Some Problems on Ruled Hypersurfaces in Nonflat Complex SpaceForms , Results Math 75, (2020). https://doi.org/10.1007/s00025-020-01294-12518] H. Reckziegel,
Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion in Global differential geometry and global analysis, Berlin, Ger-many 1984, Proceedings of a Conference held in Berlin, June 10-14, 1984. Edited by D.Ferus, R.B. Gardner, S. Helgason and U. Simon.[19] R. Takagi, On homogeneous real hypersurfaces in a complex projective space,
Osaka J.Math.10